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Slide 2 Multiplicity Codes Swastik Kopparty (Rutgers) (based on [K-Saraf-Yekhanin 11], [K 12], [K 14]) Slide 3 This talk Multiplicity codes Generalize polynomial-evaluation codes Based on evaluating polynomials & derivatives Local decoding Gave first locally decodable codes with rate 1 List-decoding Attain list-decoding capacity Beyond Local list-decoding, pseudorandom constructions, Slide 4 Error-correcting codes Codewords c r Slide 5 Local decoding/correction Slide 6 Encoding Raw data Codeword Corrupted Slide 7 Decoding Corrupted Codeword Original data What if we want to see the original data? Decoder Slide 8 What if we are only interested in one bit (or a few bits) of the original data? Slide 9 Decoding 1 bit Corrupted Codeword Message bit number i What if we want only message bit number i? Decoder Slide 10 Locally Decoding 1 bit Corrupted Codeword Original bit number i Slide 11 Locally Decoding 1 bit (slow motion) Corrupted Codeword Original bit number i Slide 12 A multivariate polynomial code Large finite field F q of size q > 0 Interpret original data as a polynomial P(X,Y) degree(P) d = (1- ) q Encoding: Evaluate P at each point of F q 2 Rate (d 2 /2)/q 2 = (1- ) 2 /2 Distance = Two low degree polynomials cannot agree on many points of F q 2 Fq2Fq2 Slide 13 Locally correcting polynomial codes Main idea: Restricting a low-degree multivariate polynomial to a line gives a low- degree univariate polynomial So: The value of P(a,b) can be recovered from the values of P on any line L through (a,b) Even if there are errors Picking the line L at random Less than /2 errors on the line (univariate polynomial decoding) Decoding queries: # points on a line = q k 0.5 Fq2Fq2 (a,b) L Slide 14 More variables Generalization: m variables Now m-variate polynomials with degree (1- ) q = d Distance = Rate (d m /m!)/q m = (1- ) m /m! Local correction via lines again Decoding time q O(k 1/m ) Slide 15 Multiplicity codes [KSY 11] Slide 16 Derivatives Given a polynomial P(X, Y) 2 F[X,Y] Define P X = dP/dX, P Y = dP/dY, P XX, P XY, Everything formally If F has small characteristic: then use Hasse derivatives Multiplicity P(X,Y) vanishes at (a,b) with mult s if for all i,j with i+j < s, P X i Y j (a,b) = 0 Notation: Mult(P, (a,b)) Mult(P, (a,b)) 1, P(a,b) = 0 Slide 17 Agreement multiplicity Defn: If all derivatives of P and Q of order < s agree on (a,b) then P and Q agree on (a,b) with multiplicity s. Mult(P-Q, (a,b)) s Slide 18 Multiplicity codes Fq2Fq2 Slide 19 Distance of multiplicity codes How many (a,b) s.t. P(a,b) = Q(a,b), P X (a,b) = Q X (a,b), P Y (a,b) = Q Y (a,b) P and Q have a multiplicity 2 agreement at (a,b) Lemma (see [Dvir-K-Saraf-Sudan 09]): Even high degree polynomials cannot have too many high-multiplicity zeroes For every P of degree at most d: E x 2 F m [ mult(P, x) ] d/q ) two polynomials P and Q cannot agree with multiplicity s in more than d/sq fraction of the points of F q 2 ) the previous multiplicity code has distance Slide 20 Locally correcting multiplicity codes Main idea: Restricting a multivariate polynomial along with its derivatives to a line gives a univariate polynomial along with its derivative As before: Pick random line L through (a,b) Looking at P, P X, P Y restricted to L is enough to give the univariate poly P| L Even if there are errors Univariate multiplicity decoding! Nielsen, Rosenbloom-Tsfasman This gives P(a,b) and dP/dL (a,b) We want P(a,b), P X (a,b), P Y (a,b) Pick another random line L dP/dL and dP/.dL are enough to give P X and P Y Fq2Fq2 (a,b) L Slide 21 Higher multiplicities Consider 2-variable polynomials of degree s(1- )q Evaluate all derivatives upto order s Rate = s/(s+1) (1- ) 2 This approaches 1 ! Decoding: s random lines through (a,b) Decoding time s q O(k 0.5 ) Slide 22 More variables, many derivatives Slide 23 L x Slide 24 Local decodability of multiplicity codes [KSY 11] Slide 25 List-Decoding Multiplicity Codes [K12] Slide 26 List-Decoding Multiplicity Codes Codewords c r Slide 27 Two theorems on list-decoding multiplicity codes Slide 28 List-Decoding Univariate Multiplicity Codes Slide 29 List-Decoding Univariate Polynomial Codes [Sudan, Guruswami-Sudan] Given r: F q F q, find f(X) of degree at most d which is close to r. Based on Interpolation + Root-finding Step 1: Interpolation: find Q(X,Y) s.t. Q(x, r(x)) = 0 for each x Claim: any f(X) which is close to r satisfies Q(X, f(X)) = 0 Step 2: Solving a polynomial equation: Solve Q(X, f(X)) = 0 (as a formal polynomial equation) Find all such f with degree at most d. (can be done by standard algorithms) Slide 30 List-Decoding Univariate Multiplicity Codes Slide 31 List-Decoding Multivariate Multiplicity Codes Slide 32 List-decoding multivariate multiplicity codes Slide 33 Other results Slide 34 Summary and Questions Codes of rate approaching 1 Decodable in O(k ) time List-decodable beyond half the minimum distance in O(k ) time Alternative construction of codes achieving list-decoding capacity For all we know, there exist codes with: Rate approaching 1 Decodable in polylog(k) time List-decoding upto capacity with constant list-size? Use multiplicity codes in practice? Already theoretically practical Further applications/generalizations?